17.7 Chebyshev Curve Fitting

The operator family \(Chebyshev\_\ldots \) implements approximation and evaluation of functions by the Chebyshev method. Let \(T_n^{(a,b)}(x)\) be the Chebyshev polynomial of order \(n\) transformed to the interval \((a,b)\). Then a function \(f(x)\) can be approximated in \((a,b)\) by a series

\[ f(x) \approx \sum _{i=0}^N c_i T_i^{(a,b)}(x) \] The operator chebyshev_fit computes this approximation and returns a list, which has as first element the sum expressed as a polynomial and as second element the sequence of Chebyshev coefficients \(c_{i}\). chebyshev_df and chebyshev_int transform a Chebyshev coefficient list into the coefficients of the corresponding derivative or integral respectively. For evaluating a Chebyshev approximation at a given point in the basic interval the operator chebyshev_eval can be used. Note that Chebyshev_eval is based on a recurrence relation which is in general more stable than a direct evaluation of the complete polynomial.

chebyshev_fit

\((fcn,var=(lo .. hi),n)\)

chebyshev_eval

\((coeffs,var=(lo .. hi),var=pt)\)

chebyshev_df

\((coeffs,var=(lo .. hi))\)

chebyshev_int

\((coeffs,var=(lo .. hi))\)

where \(\langle \)fcn\(\rangle \) is an algebraic expression (the function to be fitted), \(\langle \)var\(\rangle \) is the variable of \(\langle \)fcn\(\rangle \), \(\langle \)lo\(\rangle \) and \(\langle \)hi\(\rangle \) are numerical real values which describe an interval (\(lo < hi\)), \(\langle \)n\(\rangle \) is the approximation order, a positive integer, set to 20 if missing, \(\langle \)pt\(\rangle \) is a numerical value in the interval and \(\langle \)coeffs\(\rangle \) is a series of Chebyshev coefficients, computed by one of the operators chebyshev_coeff, chebyshev_df, or chebyshev_int.

Example:

on rounded;

w:=chebyshev_fit(sin x/x,x=(1 .. 3),5);

w := {0.0382345446975*x  - 0.239802588672*x

       + 0.0651206939005*x + 0.977836217464,

      {0.899091895826,-0.406599215895,

       -0.00519766024352,0.00946374143079,

       -0.0000948947435876}}

chebyshev_eval(second w, x=(1 .. 3), x=2.1);

0.411091086819

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